First, I'm not a pro...
Many of the drains use a compression fitting, some thread together, a few are glued. Can't tell without looking, so yes, you could break something if you just pulled it out. Might not, but probably.
If the tub has a skirt (sounds like it does), then unless you can make an access hole in the wall at the head of the tub from another room, then you'll probably have to put a hole in the ceiling from below.
Now, if you are going to throw that tub away, you could use a sawsall and just cut part of the tub skirt away, then you can get a good look at the connection. You should be able to figure it out and get the rest of the tub out. Depending on the drain for your new tub, my guess is that you'd need to cut the existing drain out (assuming it is glued in) along with the trap, then assemble a new one in the proper position. Again, though, it would be a lot easier if you had access to this connection while you set the tub in place, otherwise it is probably going to take a few install, remove cycles to line things up. Here, with limited access is where a pro (could) shine in getting it all together right. If I were going to do it, I'd measure as close as I could, dry fit things, set the tub in place, verify it fits, take the tub out mark each connection so thatyou can put it back exactly as needed, then glue things up. Problem is, you only get one chance to do this right as you glue it. That is where a pro with experience and finesse can be worth the cost. If you have access from either the end through the wall, or from underneath in the ceiling, then it is much easier to do for either a DIY or a pro.
Lonny or one of the other pros will probably give some other thoughts...just have to wait until they finish their paying jobs for the day!